Partial regularity for a Liouville system

Juan Dávila; Olivier Goubet

doi:10.3934/dcds.2014.34.2495

Article Contents

2014, Volume 34, Issue 6: 2495-2503. Doi: 10.3934/dcds.2014.34.2495

This issue Previous Article On the converse problem for the Gross-Pitaevskii equations with a large parameter Next Article Some symmetry results for entire solutions of an elliptic system arising in phase separation

Partial regularity for a Liouville system

Juan Dávila^1, and
Olivier Goubet^2,

1.
Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago
2.
LAMFA, UMR CNRS 7352, Université de Picardie Jules Verne, 33 rue St Leu, 80039, Amiens Cedex

Received: February 2013

Revised: March 2013

Published: June 2014

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Abstract

Let $\Omega\subset\mathbb{R}^n$ be a bounded smooth open set. We prove that the singular set of any extremal solution of the system \begin{equation*} -\Delta u=\mu e^v , \quad - \Delta v=\lambda e^u\quad\mbox{ in }\Omega, \end{equation*} with $u=v=0$ on $\partial \Omega$, $\mu,\lambda\geq0$, has Hausdorff dimension at most $n-10$.

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Mathematics Subject Classification: Primary: 35G30, 35B65; Secondary: 35P30.

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